\begin{tikzpicture}
[line cap=round,line join=round,x=1cm,y=1cm]
\tikzset{
        every pin/.style={fill=yellow!50!white,rectangle,rounded corners=3pt,font=\tiny},
        small dot/.style={fill=black,circle,scale=0.3},}
\begin{axis}[
x=2cm,y=2cm,
axis lines=middle,
axis line style = {-latex},
xmin=-1,
xmax=3,
ymin=-0.5,
ymax=2,
ytick={1,2},
yticklabels={1,2},
xtick=\empty,
xlabel=$x$,
ylabel=$y$,
extra x ticks={0.5},
    extra x tick style={
        tick label style={anchor=north}},
    extra x tick labels={$\frac{1}{2}$},
enlargelimits = true,
]



\draw [thick, dashed] (axis cs: -0.05,1) -- (axis cs: 0.5,1);
\draw [thick, dashed] (axis cs: 0.5,-0.05) -- (axis cs: 0.5,1);


\addplot[domain=-1:0.5, blue, line width=1,samples=2000] {((x^2)-(1/4))/(x-(1/2)};
\addplot[color=blue!80!black, only marks, style={mark=*, fill=white}] coordinates {(0.5,1)};
\addplot[domain=0.51:1.5, blue, line width=1,samples=2000] {((x^2)-(1/4))/(x-(1/2)};

\addplot[color=blue!80!black, only marks, style={mark=*}] coordinates {(0.5,2)};

\draw [thick, dashed] (axis cs: -0.05,2) -- (axis cs: 0.5,2);
\draw [thick, dashed] (axis cs: 0.5,1) -- (axis cs: 0.5,2);



\scriptsize{ \node () at (axis cs:2.4,1.8) {$\displaystyle y=\left\{ \begin{array}{rl}
\frac{x^2-\frac{1}{4}}{x-\frac{1}{2}} & \text{für}\; x \neq \frac{1}{2}\\ \\
2 & \text{für}\; x = \frac{1}{2}\\
\end{array}
\right .$};}

% \draw[] (axis cs:1.3, 0.1) -- (axis cs:1.3, -0.1);
% \node () at (axis cs:1.3,-0.25) {$x_0$};

% \draw[] (axis cs:1.14,-0.05) -- (axis cs:1.14, 0.05);
% \node () at (axis cs:0.33,-0.45) {$\frac{1}{\pi}$};

\end{axis}


\end{tikzpicture}